# Facility location problems

Author: Aaron Litoff
Stewards: Dajun Yue and Fengqi You

MILP [1]

The facility location problem deals with selecting the placement of a facility (often from a list of integer possibilities) to best meet the demanded constraints. The problem consists of finding the location to build a factory that minimizes total weighted distances from suppliers and customers, where weights are representative of the difficulty of transporting materials. The solution to this problem gives the highest profit choice that most efficiently serves the needs of all customers.

## History

The Fermat-Weber problem was one of the first facility location problems every proposed, and was done so as early as the 17th century. This "geometric median of three points" can be thought of as a version of the facility location problem where the assumption is made that transportation costs per distance are the same for all destinations. It was put forth by the French mathematician Pierre de Fermat to the Italian physicist Evangelista Torricelli as follows:

"Given three points in a plane, find a fourth point such that the sum of its distances to the three given points is as small as possible." [1]

In 1909 Alfred Weber used a three point version to model possible industrial locations in order to minimize transportation costs from two sources to a single customer. [2] This is one the simplest continuous facility location models.

The Fermat Point, or Torricelli Point, is the solution that minimizes distances from A, B, and C (the three corners of the black triangle).[3]

A modern day engineering interpretation could be as follows:
"Find the best location for a refining plant between three cities in such a way that the sum of the connections between the plant and the cities in minimal."

## Description and Formulation

Binary decision variables

## Examples and Applications

### A Real World Example

Suppose you are a manager at a company that builds Warehouse needs to be built in a central location so that the transportation costs are minimized

Applications in physics, solution to Fermat-Weber problem allowing for "repulsion" or negative distances.

## References

[1] Dorrie, H. (1965) 100 great problems of elementary mathematics: their history and solution. New York, NY: Dover Publications.

[2] Drezner, Z & Hamacher. H. W. (2004). Facility location: applications and theory. New York, NY: Springer.

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